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MIT_Aspirant
 
 

If 2^x - 2^x-2 = 3*(2)^13, What is the value of x?

by MIT_Aspirant Mon Aug 25, 2008 9:51 pm

If 2^x - 2^x-2 = 3*(2)^13, What is the value of x?

This is from the GMAT Prep 1. Can someone please walk me through this?
Raj
 
 

Re: If 2^x - 2^x-2 = 3*(2)^13, What is the value of x?

by Raj Tue Aug 26, 2008 12:30 am

LHS:
2^x(1- 1/4) = 2^x(3/2^2) = 3*2^(x-4). Now equate to RHS 3*2^(x-4) = 3*(2)^13, which means x-4 = 13, x=17.

-Raj.

MIT_Aspirant wrote:If 2^x - 2^x-2 = 3*(2)^13, What is the value of x?

This is from the GMAT Prep 1. Can someone please walk me through this?
guest
 
 

the first conversion

by guest Tue Aug 26, 2008 7:53 am

can you explain how you converted 2^x - 2^x-2 into 2^x(1- 1/4)?
MIT_Aspirant
 
 

by MIT_Aspirant Tue Aug 26, 2008 8:00 am

Hi Raj,

The correct answer is 15 not 17.
Anand
 
 

Solution to the prob.

by Anand Tue Aug 26, 2008 11:43 am

Hi MIT_Aspirant,

This is how I approached this prob.

2^(X) - 2^(X-2) = 3* 2^(13)
Taking 2^(X) common on the left side, we get:
2^(X)[1- 2^(-2)] = 3* 2^(13)
2^(X)[1- (1/4)] = 3* 2^(13) ...Since 2^(-2)=(1/4)
2^(X)[(3/4)] = 3* 2^(13)
2^(X) = 3* 2^(13) * (4/3)
2^(X) = 2^(13+2) ...Since we cancel 3’s and 4=2^(2)
X=15

Thanks,
Anand
Raj
 
 

Re: If 2^x - 2^x-2 = 3*(2)^13, What is the value of x?

by Raj Tue Aug 26, 2008 1:56 pm

Sorry about that.. 2^x(3/2^2) = 3*2^(x-4) is NOT right. It should say 2^x(3/2^2) = 3*2^(x-2), hence x-2 =17, x = 15.

I hope not to make such mistakes in the actual exam !!
-Raj.

Raj wrote:LHS:
2^x(1- 1/4) = 2^x(3/2^2) = 3*2^(x-4). Now equate to RHS 3*2^(x-4) = 3*(2)^13, which means x-4 = 13, x=17.

-Raj.

MIT_Aspirant wrote:If 2^x - 2^x-2 = 3*(2)^13, What is the value of x?

This is from the GMAT Prep 1. Can someone please walk me through this?
RonPurewal
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by RonPurewal Sun Sep 07, 2008 1:39 am

ok guys, your opener works here, but there's an easier way.
remember - you should make connections between similar-looking problems. and this problem is similar to just about any other problem in which you're factoring a common power out of a polynomial.

here's the deal: when you factor a power out of a polynomial, which power do you factor out: the smallest common power, or the biggest common power?
that's right, the smallest one. if you have x^5 - x^2, you only factor out x^2, not x^5.
analogy:
on this problem, just factor out 2^(x - 2).
this gives
left hand side = [2^(x - 2)](2^2 - 1)
= [2^(x - 2)](3)
you can then cancel the 3's, leaving 2^(x - 2) = 2^13. therefore, x - 2 = 13, so x = 15.
done.

btw, the most important part of this post is the 'make analogies' part. if you focus on the parts of each problem that remind you of other problems, then you're going to face a lot less memorization and a lot more progress.
Priyanka
 
 

by Priyanka Mon Sep 22, 2008 1:00 pm

Instead of focusing on the LHS , we can try and simplify the RHS

LHS = 3 * 2^13

can be written as (2^2 - 1) * 2^13.

= 2^15 - 2^13.

now RHS = 2^x - 2^(x-2) = LHS = 2^15 - 2^13

therefore x = 15.
RonPurewal
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by RonPurewal Mon Oct 13, 2008 7:31 am

Priyanka wrote:Instead of focusing on the LHS , we can try and simplify the RHS

LHS = 3 * 2^13

can be written as (2^2 - 1) * 2^13.

= 2^15 - 2^13.

now RHS = 2^x - 2^(x-2) = LHS = 2^15 - 2^13

therefore x = 15.


this is pretty neat, although it's not algorithmic; i.e., it's a nice little heuristic that will shave some time off the problem if it happens to work, but there's no guarantee that it will work. in other situations, it could crash and burn.
on this particular problem, though, it's beautiful.

factoring approaches, on the other hand, are usually algorithmic; i.e., they'll work, regardless of the nature of the problem.
benkriger
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Re: If 2^x - 2^x-2 = 3*(2)^13, What is the value of x?

by benkriger Fri Oct 16, 2009 5:45 pm

2x - 2^(x-2) = 3(2^13), what is x?

1) We can see from the statement that we are going to be working with the number 2 raised to a power. This is important, because there are patterns when taking the number 2 to a power.

2) Lets examine the LHS 2^x - 2^(x-2) by picking numbers and looking for a pattern.

-If we let x equal 4, then we get (2^4) - (2^2) = (16)-(4) = 12
-Lets do one more number just to make sure the pattern holds. Let x equal 5. Then we have (2^5) - (2^3) = (32)-(8) = 24
-Now what do we realize about the numbers 12 and 24? Well at first look you may not be sure. But looking at the RHS of the equation, you can see that they represented the right side as three times the number two to a power. So lets try breaking down each of these numbers into that pattern. 12=3*4 and we can break that down into 12= 3*2*2 and then we can simplify that into 12 =3*2^2. Do the same thing with the 24=3*8 which also means 24=3*2*2*2 and then finally 24=3*2^3.

What do we notice? If x=4, our equation was 3*2^2 and if x=5, we had 3*2^3. Both of these equations is simply 3*2^(x-2).

3)

We we now rearranged the equation to: 3*2^(x-2)=3*2^13 and now set x-2=13 and solve for x, and X=15.
RonPurewal
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Re: If 2^x - 2^x-2 = 3*(2)^13, What is the value of x?

by RonPurewal Sat Oct 24, 2009 8:07 am

benkriger wrote:2x - 2^(x-2) = 3(2^13), what is x?

1) We can see from the statement that we are going to be working with the number 2 raised to a power. This is important, because there are patterns when taking the number 2 to a power.

2) Lets examine the LHS 2^x - 2^(x-2) by picking numbers and looking for a pattern.

-If we let x equal 4, then we get (2^4) - (2^2) = (16)-(4) = 12
-Lets do one more number just to make sure the pattern holds. Let x equal 5. Then we have (2^5) - (2^3) = (32)-(8) = 24
-Now what do we realize about the numbers 12 and 24? Well at first look you may not be sure. But looking at the RHS of the equation, you can see that they represented the right side as three times the number two to a power. So lets try breaking down each of these numbers into that pattern. 12=3*4 and we can break that down into 12= 3*2*2 and then we can simplify that into 12 =3*2^2. Do the same thing with the 24=3*8 which also means 24=3*2*2*2 and then finally 24=3*2^3.

What do we notice? If x=4, our equation was 3*2^2 and if x=5, we had 3*2^3. Both of these equations is simply 3*2^(x-2).

3)

We we now rearranged the equation to: 3*2^(x-2)=3*2^13 and now set x-2=13 and solve for x, and X=15.



yes, this is valid pattern recognition. although it takes an extremely strong numerical intuition to get this from just seeing 12 and 24.
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Re:

by abdt_80 Thu Aug 19, 2010 6:08 pm

Priyanka wrote:Instead of focusing on the LHS , we can try and simplify the RHS

LHS = 3 * 2^13

can be written as (2^2 - 1) * 2^13.

= 2^15 - 2^13.

now RHS = 2^x - 2^(x-2) = LHS = 2^15 - 2^13

therefore x = 15.


Thanks for making easy.
RonPurewal
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Re: Re:

by RonPurewal Thu Sep 16, 2010 6:25 am

there's a nice collection of solutions on this page.
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Re: If 2^x - 2^x-2 = 3*(2)^13, What is the value of x?

by ehassen Thu Sep 23, 2010 5:06 pm

Image
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Re: If 2^x - 2^x-2 = 3*(2)^13, What is the value of x?

by RonPurewal Mon Oct 04, 2010 8:41 am

also, don't forget the method of plugging in the answer choices and working backward.

start in the middle (choice c). if you try that choice, you get
2^13 - 2^11 = 3(2^13)
this definitely doesn't work -- it says that if you take a number and subtract something from it, then you wind up with three times the original number. that's impossible.
also, it's clear that we must make x bigger in order for this to work.

try the next bigger choice (d):
2^15 - 2^13 = 3(2^13)
factor the left-hand side:
2^13(2^2 - 1) = 3(2^13)
so that's a true statement.
choice (d) wins

--

also, if you really back into a corner on this one, you can always just work out the powers of 2.
especially if you use the estimate that 2^10 is approximately 1000, you can quite easily work out the correct answer to this problem using raw arithmetic, well within the given time constraint.
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